Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-6x+6y &= -6 \\ 7x-5y &= -3\end{align*}$
Answer: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $5$ and the bottom equation by $6$ $\begin{align*}-30x+30y &= -30\\ 42x-30y &= -18\end{align*}$ Add the top and bottom equations. $12x = -48$ Divide both sides by $12$ and reduce as necessary. $x = -4$ Substitute $-4$ for $x$ in the top equation. $-6( -4)+6y = -6$ $24+6y = -6$ $6y = -30$ $y = -5$ The solution is $\enspace x = -4, \enspace y = -5$.